Question

The roots of the quadratic equation $x^2 + 5x + 6 = 0$ will be :

• A. -3, -2
• B. 3, 2
• C. -3, 2
• D. 3, -2

Hint:

If all coefficients in the quadratic equation are positive, the roots will be negative.

Answer 1

The Correct Option is A

To determine the roots of the quadratic equation \(x^2 + 5x + 6 = 0\), we will use the factorization method.

1. Compare the equation \(x^2 + 5x + 6 = 0\) with the general quadratic equation \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = 5\), and \(c = 6\).
2. To factorize the quadratic expression, we need two numbers whose product is \(a \cdot c = 1 \cdot 6 = 6\) and whose sum is \(b = 5\).
3. The numbers that satisfy these conditions are \(2\) and \(3\) because:
   • Product: \(2 \cdot 3 = 6\)
   • Sum: \(2 + 3 = 5\)
4. Now, rewrite the middle term \(5x\) as \(2x + 3x\):
   • \(x^2 + 5x + 6 = x^2 + 2x + 3x + 6\)
5. Factor by grouping:
   • Group the first two terms: \(x(x + 2)\)
   • Group the last two terms: \(3(x + 2)\)
6. Combine these to get: \((x + 2)(x + 3) = 0\)
7. Setting each factor to zero gives the roots:
   • \(x + 2 = 0 \implies x = -2\)
   • \(x + 3 = 0 \implies x = -3\)

The roots of the quadratic equation \(x^2 + 5x + 6 = 0\) are -3 and -2.

Therefore, the correct answer is: -3, -2.